22.1.29 problem 29

Internal problem ID [4335]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 29
Date solved : Tuesday, September 30, 2025 at 07:20:28 AM
CAS classification : [[_homogeneous, `class D`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} y \left (2 x -y+2\right )+2 \left (x -y\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 64

ode:=y(x)*(2*x-y(x)+2)+2*(x-y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {{\mathrm e}^{-x} \sqrt {{\mathrm e}^{x} c_1 \left ({\mathrm e}^{x} c_1 \,x^{2}+1\right )}+c_1 x}{c_1} \\ y &= \frac {-{\mathrm e}^{-x} \sqrt {{\mathrm e}^{x} c_1 \left ({\mathrm e}^{x} c_1 \,x^{2}+1\right )}+c_1 x}{c_1} \\ \end{align*}

✓ Mathematica. Time used: 40.194 (sec). Leaf size: 125

ode=y[x]*(2*x-y[x]+2)+2*(x-y[x])*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x-e^{-x} \sqrt {e^x \left (e^x x^2-e^{2 c_1}\right )}\\ y(x)&\to x+e^{-x} \sqrt {e^x \left (e^x x^2-e^{2 c_1}\right )}\\ y(x)&\to x-e^{-x} \sqrt {e^{2 x} x^2}\\ y(x)&\to e^{-x} \sqrt {e^{2 x} x^2}+x \end{align*}

✓ Sympy. Time used: 3.225 (sec). Leaf size: 82

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((2*x - 2*y(x))*Derivative(y(x), x) + (2*x - y(x) + 2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = x - \sqrt {x^{2} - \sqrt {C_{1} e^{- 2 x}}}, \ y{\left (x \right )} = x + \sqrt {x^{2} - \sqrt {C_{1} e^{- 2 x}}}, \ y{\left (x \right )} = x - \sqrt {x^{2} + \sqrt {C_{1} e^{- 2 x}}}, \ y{\left (x \right )} = x + \sqrt {x^{2} + \sqrt {C_{1} e^{- 2 x}}}\right ] \]